1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions

## Cauchy Distribution

Probability Density Function The general formula for the probability density function of the Cauchy distribution is

$$f(x) = \frac{1} {s\pi(1 + ((x - t)/s)^{2})}$$

where t is the location parameter and s is the scale parameter. The case where t = 0 and s = 1 is called the standard Cauchy distribution. The equation for the standard Cauchy distribution reduces to

$$f(x) = \frac{1} {\pi(1 + x^{2})}$$

Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.

The following is the plot of the standard Cauchy probability density function.

Cumulative Distribution Function The formula for the cumulative distribution function for the Cauchy distribution is

$$F(x) = 0.5 + \frac{\arctan{(x)}} {\pi}$$

The following is the plot of the Cauchy cumulative distribution function.

Percent Point Function The formula for the percent point function of the Cauchy distribution is

$$G(p) = -\cot{(\pi p)}$$

The following is the plot of the Cauchy percent point function.

Hazard Function The formula for the Cauchy hazard function is

$$h(x) = \frac{1} {(1 + x^2)(0.5 \pi - \arctan{x})}$$

The following is the plot of the Cauchy hazard function.

Cumulative Hazard Function The formula for the Cauchy cumulative hazard function is

$$H(x) = -\ln \left( 0.5 - \frac{\arctan{x}}{\pi} \right)$$

The following is the plot of the Cauchy cumulative hazard function.

Survival Function The formula for the Cauchy survival function is

$$S(x) = 0.5 - \frac{\arctan{(x)}} {\pi}$$

The following is the plot of the Cauchy survival function.

Inverse Survival Function The formula for the Cauchy inverse survival function is

$$Z(p) = -\cot{(\pi (1 - p))}$$

The following is the plot of the Cauchy inverse survival function.

Common Statistics
 Mean The mean is undefined. Median The location parameter t. Mode The location parameter t. Range $$-\infty \mbox{ to } \infty$$ Standard Deviation The standard deviation is undefined. Coefficient of Variation The coefficient of variation is undefined. Skewness The skewness is 0. Kurtosis The kurtosis is undefined.
Parameter Estimation The likelihood functions for the Cauchy maximum likelihood estimates are given in chapter 16 of Johnson, Kotz, and Balakrishnan. These equations typically must be solved numerically on a computer.
Comments The Cauchy distribution is important as an example of a pathological case. Cauchy distributions look similar to a normal distribution. However, they have much heavier tails. When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of how sensitive the tests are to heavy-tail departures from normality. Likewise, it is a good check for robust techniques that are designed to work well under a wide variety of distributional assumptions.

The mean and standard deviation of the Cauchy distribution are undefined. The practical meaning of this is that collecting 1,000 data points gives no more accurate an estimate of the mean and standard deviation than does a single point.

Software Many general purpose statistical software programs support at least some of the probability functions for the Cauchy distribution.